1. (3 marks) Answer the following questions related to POPULATION, SAMPLE and VARIABLES:
(a) (1.5 marks) Ten students enrolled at Griffith University were surveyed regarding A: number of courses enrolled in; B: total cost of textbooks; and C: method of payment used for textbooks. Identify the population, the sample, and classify the three variables as either qualitative or quantitative.
(b) (1.5 marks) A quality-control inspector selects assembled parts from an assembly line and records the information concerning each part as: A: defective or non-defective; B: the employee number of the individual who assembled the part; C: the weight of the part. Identify the population, the sample, and classify the three variables as either qualitative or quantitative.
2. (2.5 marks) Identify the statements below as TRUE or FALSE. If FALSE, explain.
(a) A volunteer sample consists of results collected from elements of the population that chose to contribute the needed information on their own initiative.
(b) Variability is the extent to which data points in a statistical distribution or data set diverge from the standard deviation.
(c) Random sampling is a method of selecting a sample from a statistical population in such a way that every possible sample that could be selected has a same predetermined probability of being selected.
(d) A biased sampling method is a sampling method that produces data that are representative of the sampled population.
(e) A convenience sample occurs when some elements of the population choose to contribute the needed information on their own initiative.
3. (10 marks) A certain polymer is used for evacuation systems for aircraft. It is important that the polymer be resistant to the aging process. Twenty specimens of the polymer were used in an experiment. Ten were assigned randomly to be exposed to an accelerated batch aging process that involved exposure to high temperatures for 10 days. Measurements of tensile strength of the specimens were made, and the following data were recorded on tensile strength in psi:
(a) (2 marks) Show a dot plot of the data with both “No aging” and “Aging” tensile strength values in the same graph.
(b) (2 marks) Based on the plot, does it appear as if the aging process has had an effect on the tensile strength of this polymer? Comment further.
(c) (1 mark) Calculate the sample mean tensile strength of the two samples, to one decimal digit.
(d) (3 marks) Calculate the median for both samples, to one decimal digit.
(e) (2 marks) Discuss the similarity or the lack of similarity between the mean and median in each group.
4. (12 marks) The following data are the mean daily water levels (m) recorded by two submersible pressure sensors manufactured by two different companies (A and B). The measurements were taken at the Nerang River (QLD), at a point located 15 km upstream of the river mouth, over a period of 10 days.
a) (3 marks) For each pressure sensor, compute the sample mean, median, mode and midrange (to two decimal digits).
b) (6 marks) For each sensor, compute the variance and standard deviation of the data (to two decimal digits).
c) (3 marks) Knowing that the mean water level at the Nerang River was 5.8 m in that period of time, which sensor would you select to measure river water level and why?
5. (12 marks) The following frequency distribution provides the number of engineering managers and their annual salaries (in $1000):
(a) (2 marks) Prepare a cumulative relative frequency distribution table for this frequency distribution.
(b) (3 marks) Estimate the mean annual salary from the grouped data (to one decimal digit).
(c) (4 marks) Construct two graphs: a histogram and an ogive
(d) (3 marks) The actual mean (calculated using the individual, original set of data) of the annual salaries of these 200 managers is $35,700. Explain why the estimated mean from the grouped data (part b) is different from the actual mean.
6. (12 marks) The annual salaries (in $100) of high school teachers employed at one of the public high schools in Kent County, Michigan, are listed below:
(a) (2 marks) Find the first quartile for the above data and interpret the result.
(b) (2 marks) Find the third quartile for the above data and interpret the result.
(c) (2 marks) Find the median of the data and interpret the results.
(d) (2 marks) What is the interquartile range of the data and what does it mean?
(e) (4 marks) Represent the data using a ‘box-and-whiskers’ plot, displaying the 5 number summary of the data on the graph.
7. (16 marks) Raw material used in the production of a synthetic fibre is stored in a room that has no humidity control. Measurements of the relative humidity (%) in the storage room and the moisture content of a sample of the raw material (%) on 12 days, yielded the following results:
(a) (8 marks) Plot the data on a scatter plot. Based on this data set, is there a linear correlation between relative humidity and moisture content in the raw material? Back up your argument with the calculation of the coefficient of linear correlation (r). You must clearly show all steps used in your calculations. Round numbers to two decimal places.
(b) (4 marks) Through regression analysis, find the equation of the line that best describes the relationship between the two variables. You must clearly show all steps used in your calculations.
(c) (1.5 marks) Calculate the coefficient of determination, r2, and describe in your own words what it represents.
(d) (2.5 marks) What would be an estimated value of moisture content when the relative humidity in the storage room is 50%? What about when the relative humidity is 80%?
8. (8 marks) Consider a family of three children.
(a) (4 marks) Use a tree diagram to show all of the possible outcomes.
(b) (4 marks) Find the probability that this family has at least one child of each gender.
9. (5 marks) A certain type of storage battery lasts, on average, 3.5 years with a standard deviation of 0.5 year. Assuming that battery life is normally distributed, find the probability that a given battery will last more than 2.5 years.
10. (4.5 marks) The machines used to fill detergent bottles in a detergent industry do not always fill the bottles to specification. In this industry, a machine may: A, fill to specification; B, underfill, or C, overfill. Let P(B) = 0.001 while P(A) = 0.990.
(a) (1.5 marks) Give P(C).
(b) (1.5 marks) What is the probability that the machine does not underfill?
(c) (1.5 marks) What is the probability that the machine either overfills or underfills?
11. (5 marks) An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If a sample of 30 bulbs has an average life of 780 hours, find a 96% confidence interval for the population mean of all bulbs produced by this firm.
12. (10 marks) A trucking firm is suspicious of the claim that the average lifetime of a certain tire is at least 28,000 km, with a standard deviation of 1,348 km. To check this claim, the trucking firm selected 40 tires to test and found a mean lifetime of 27,463 km. Use a 0.01 level of significance to test the claim that μ ≥ 28,000 km (H0) against the alternative that μ < 28,000 km (H1).